How do L2 norm and The Jacobian act as a regularisation term to encourage smoothness in a deformation field? from the VoxelMorph original paper (here) they used Jacobian as a means to smoothen the deformation field, a similar paper (here) made use of L2 norm to smoothen the deformation field. However, I don't understand how this is achieved using either L2 norm or Jacobian. Thanks in advance.
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TL,DR Use the Jacobian to get smoothness and the L2 norm to limit the magnitude + they could be complementary.
I just skimmed at the paper (that you haven't really referenced) and the regularization term using the derivatives makes sense to make $\phi$ smooth. It's basically penalizing large changes between neighboring values of the registration field. Using the L2 norm would penalize the magnitude of the registration field, not the local variations.
Franco Marchesoni
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That means taking the jacobian(derivatives, partial derivatives) measures the smoothness and then applying a penalty to the magnitude of the derivatives using L2 norm? is that what the statement "We use the L2 norm over the deformation field gradients as a regularization term to encourage the deformation field smoothness" means (I've included the links sorry about that) – a__ys Dec 03 '22 at 20:50
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Exactly, if the gradient or changes are small, the field is smooth. – Franco Marchesoni Dec 05 '22 at 17:22